Negative binomial construction of random discrete distributions on the infinite simplex

Yuguang F. Ipsen, Ross A. Maller
Theory of Stochastic Processes
Vol.22 (38), no.2, 2017, pp.34-46
The Poisson-Kingman distributions, PK(ρ), on the infinite simplex, can be constructed from a Poisson point process having intensity density ρ or by taking the ranked jumps up till a specified time t>0 of a subordinator with Lévy density ρ, as proportions of the subordinator. As a natural extension, we replace the Poisson point process with a negative binomial point process having parameter r>0 and Lévy density ρ, thereby defining a new class PK^(r)(ρ) of distributions on the infinite simplex. The new class contains the two-parameter generalisation PD(α, θ) of [13] when θ>0. It also contains a class of distributions derived from the trimmed stable subordinator. We derive properties of the new distributions, with particular reference to the two most well-known PK distributions: the Poisson-Dirichlet distribution PK(ρ_θ) generated by a Gamma process with Lévy density ρ_θ(x) = θ e^-x/x, x>0, θ > 0, and the random discrete distribution, PD(α,0), derived from an α-stable subordinator.